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I made up a problem and wanted to make sure that I did everything right.
A thing rod with mass M is bent into the shape of a quarter circle with radius R. It is positioned so that it looks like the arc made as theta moves from terminal position (+ x axis) to 90 degrees counterclockwise. I am going to let this arc rotate freely about the x-axis. My problem is to determine the speed of the tip of the wire when it has rotated all the way to the bottom.
First I calculated the potential energy of the rod using the x-axis as a refrence frame. I did this by integrating over the rod noting that the height of each piece is Rsin@
U=(2/pi)MgR.
Then I calculated the moment of inertia using the same integration techniques.
I=(1/2)MR^2
(Is that really true!?!?!)
Finally I doubled the initial potential energy to get the kinetic when it has spun around and set that equal to (1/2)Iw^2
v=4[gR/pi]^(1/2)
If anyone would check this I would be very appreciative.
A thing rod with mass M is bent into the shape of a quarter circle with radius R. It is positioned so that it looks like the arc made as theta moves from terminal position (+ x axis) to 90 degrees counterclockwise. I am going to let this arc rotate freely about the x-axis. My problem is to determine the speed of the tip of the wire when it has rotated all the way to the bottom.
First I calculated the potential energy of the rod using the x-axis as a refrence frame. I did this by integrating over the rod noting that the height of each piece is Rsin@
U=(2/pi)MgR.
Then I calculated the moment of inertia using the same integration techniques.
I=(1/2)MR^2
(Is that really true!?!?!)
Finally I doubled the initial potential energy to get the kinetic when it has spun around and set that equal to (1/2)Iw^2
v=4[gR/pi]^(1/2)
If anyone would check this I would be very appreciative.
